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On sufficient conditions for degrees of freedom counting of multi-field generalised Proca theories
Published 27 Mar 2023 in hep-th and gr-qc | (2303.15261v2)
Abstract: We derive sufficient conditions for theories consisting of multiple vector fields, which could also couple to external fields, to be multi-field generalised Proca theories. The conditions are derived by demanding that the theories have the required structure of constraints, giving the correct number of degrees of freedom. The Faddeev-Jackiw constraint analysis is used and is cross-checked by Lagrangian constraint analysis. To ensure the theory is constraint, we impose a standard special form of Hessian matrix. The derivation benefits from the realisation that the theories are diffeomorphism invariance (or, in the case of flat spacetime, invariant under Lorentz isometry). The sufficient conditions obtained include a refinement of secondary-constraint enforcing relations derived previously in literature, as well as a condition which ensures that the iteration process of constraint analysis terminates. Some examples of theories are analysed to show whether they satisfy the sufficient conditions. Most notably, due to the obtained refinement on some of the conditions, some theories which are previously interpreted as being undesirable are in fact legitimate, and vice versa. This in turn affects the previous interpretations of cosmological implications which should later be reinvestigated.
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D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Weinberg, S.: A Model of Leptons. Phys. Rev. Lett. 19, 1264–1266 (1967) https://doi.org/10.1103/PhysRevLett.19.1264 Higgs [1964a] Higgs, P.W.: Broken symmetries, massless particles and gauge fields. Phys. Lett. 12, 132–133 (1964) https://doi.org/10.1016/0031-9163(64)91136-9 Higgs [1964b] Higgs, P.W.: Broken Symmetries and the Masses of Gauge Bosons. Phys. Rev. Lett. 13, 508–509 (1964) https://doi.org/10.1103/PhysRevLett.13.508 Englert and Brout [1964] Englert, F., Brout, R.: Broken Symmetry and the Mass of Gauge Vector Mesons. Phys. Rev. Lett. 13, 321–323 (1964) https://doi.org/10.1103/PhysRevLett.13.321 Guralnik et al. [1964] Guralnik, G.S., Hagen, C.R., Kibble, T.W.B.: Global Conservation Laws and Massless Particles. Phys. Rev. Lett. 13, 585–587 (1964) https://doi.org/10.1103/PhysRevLett.13.585 Guth [1981] Guth, A.H.: The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems. Phys. Rev. D 23, 347–356 (1981) https://doi.org/10.1103/PhysRevD.23.347 Perlmutter et al. [1999] Perlmutter, S., et al.: Measurements of ΩΩ\Omegaroman_Ω and ΛΛ\Lambdaroman_Λ from 42 high redshift supernovae. Astrophys. J. 517, 565–586 (1999) https://doi.org/10.1086/307221 arXiv:astro-ph/9812133 Riess et al. [1998] Riess, A.G., et al.: Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J. 116, 1009–1038 (1998) https://doi.org/10.1086/300499 arXiv:astro-ph/9805201 Rodríguez and Navarro [2018] Rodríguez, Y., Navarro, A.A.: Non-Abelian S𝑆Sitalic_S-term dark energy and inflation. Phys. Dark Univ. 19, 129–136 (2018) https://doi.org/10.1016/j.dark.2018.01.003 arXiv:1711.01935 [gr-qc] Gómez and Rodríguez [2019] Gómez, L.G., Rodríguez, Y.: Stability Conditions in the Generalized SU(2) Proca Theory. Phys. Rev. D 100(8), 084048 (2019) https://doi.org/10.1103/PhysRevD.100.084048 arXiv:1907.07961 [gr-qc] Garnica et al. [2022] Garnica, J.C., Gomez, L.G., Navarro, A.A., Rodriguez, Y.: Constant-Roll Inflation in the Generalized SU(2) Proca Theory. Annalen Phys. 534(2), 2100453 (2022) https://doi.org/10.1002/andp.202100453 arXiv:2109.10154 [gr-qc] Proca [1936] Proca, A.: Sur la theorie ondulatoire des electrons positifs et negatifs. J. Phys. Radium 7, 347–353 (1936) https://doi.org/10.1051/jphysrad:0193600708034700 Born and Infeld [1934] Born, M., Infeld, L.: Foundations of the new field theory. Proc. Roy. Soc. Lond. A 144(852), 425–451 (1934) https://doi.org/10.1098/rspa.1934.0059 Dirac [1962] Dirac, P.A.M.: An Extensible model of the electron. Proc. Roy. Soc. Lond. A 268, 57–67 (1962) https://doi.org/10.1098/rspa.1962.0124 Tasinato [2014] Tasinato, G.: Cosmic Acceleration from Abelian Symmetry Breaking. JHEP 04, 067 (2014) https://doi.org/10.1007/JHEP04(2014)067 arXiv:1402.6450 [hep-th] Heisenberg [2014] Heisenberg, L.: Generalization of the Proca Action. JCAP 05, 015 (2014) https://doi.org/10.1088/1475-7516/2014/05/015 arXiv:1402.7026 [hep-th] Hull et al. [2016] Hull, M., Koyama, K., Tasinato, G.: Covariantized vector Galileons. Phys. Rev. D 93(6), 064012 (2016) https://doi.org/10.1103/PhysRevD.93.064012 arXiv:1510.07029 [hep-th] Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized Proca action for an Abelian vector field. JCAP 02, 004 (2016) https://doi.org/10.1088/1475-7516/2016/02/004 arXiv:1511.03101 [hep-th] Beltran Jimenez and Heisenberg [2016] Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Higgs, P.W.: Broken symmetries, massless particles and gauge fields. Phys. Lett. 12, 132–133 (1964) https://doi.org/10.1016/0031-9163(64)91136-9 Higgs [1964b] Higgs, P.W.: Broken Symmetries and the Masses of Gauge Bosons. Phys. Rev. Lett. 13, 508–509 (1964) https://doi.org/10.1103/PhysRevLett.13.508 Englert and Brout [1964] Englert, F., Brout, R.: Broken Symmetry and the Mass of Gauge Vector Mesons. Phys. Rev. Lett. 13, 321–323 (1964) https://doi.org/10.1103/PhysRevLett.13.321 Guralnik et al. [1964] Guralnik, G.S., Hagen, C.R., Kibble, T.W.B.: Global Conservation Laws and Massless Particles. Phys. Rev. Lett. 13, 585–587 (1964) https://doi.org/10.1103/PhysRevLett.13.585 Guth [1981] Guth, A.H.: The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems. Phys. Rev. D 23, 347–356 (1981) https://doi.org/10.1103/PhysRevD.23.347 Perlmutter et al. [1999] Perlmutter, S., et al.: Measurements of ΩΩ\Omegaroman_Ω and ΛΛ\Lambdaroman_Λ from 42 high redshift supernovae. Astrophys. J. 517, 565–586 (1999) https://doi.org/10.1086/307221 arXiv:astro-ph/9812133 Riess et al. [1998] Riess, A.G., et al.: Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J. 116, 1009–1038 (1998) https://doi.org/10.1086/300499 arXiv:astro-ph/9805201 Rodríguez and Navarro [2018] Rodríguez, Y., Navarro, A.A.: Non-Abelian S𝑆Sitalic_S-term dark energy and inflation. Phys. Dark Univ. 19, 129–136 (2018) https://doi.org/10.1016/j.dark.2018.01.003 arXiv:1711.01935 [gr-qc] Gómez and Rodríguez [2019] Gómez, L.G., Rodríguez, Y.: Stability Conditions in the Generalized SU(2) Proca Theory. Phys. Rev. D 100(8), 084048 (2019) https://doi.org/10.1103/PhysRevD.100.084048 arXiv:1907.07961 [gr-qc] Garnica et al. [2022] Garnica, J.C., Gomez, L.G., Navarro, A.A., Rodriguez, Y.: Constant-Roll Inflation in the Generalized SU(2) Proca Theory. Annalen Phys. 534(2), 2100453 (2022) https://doi.org/10.1002/andp.202100453 arXiv:2109.10154 [gr-qc] Proca [1936] Proca, A.: Sur la theorie ondulatoire des electrons positifs et negatifs. J. Phys. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. 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Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. 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[2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized Proca action for an Abelian vector field. JCAP 02, 004 (2016) https://doi.org/10.1088/1475-7516/2016/02/004 arXiv:1511.03101 [hep-th] Beltran Jimenez and Heisenberg [2016] Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. 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[2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Riess, A.G., et al.: Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J. 116, 1009–1038 (1998) https://doi.org/10.1086/300499 arXiv:astro-ph/9805201 Rodríguez and Navarro [2018] Rodríguez, Y., Navarro, A.A.: Non-Abelian S𝑆Sitalic_S-term dark energy and inflation. Phys. Dark Univ. 19, 129–136 (2018) https://doi.org/10.1016/j.dark.2018.01.003 arXiv:1711.01935 [gr-qc] Gómez and Rodríguez [2019] Gómez, L.G., Rodríguez, Y.: Stability Conditions in the Generalized SU(2) Proca Theory. Phys. Rev. D 100(8), 084048 (2019) https://doi.org/10.1103/PhysRevD.100.084048 arXiv:1907.07961 [gr-qc] Garnica et al. [2022] Garnica, J.C., Gomez, L.G., Navarro, A.A., Rodriguez, Y.: Constant-Roll Inflation in the Generalized SU(2) Proca Theory. Annalen Phys. 534(2), 2100453 (2022) https://doi.org/10.1002/andp.202100453 arXiv:2109.10154 [gr-qc] Proca [1936] Proca, A.: Sur la theorie ondulatoire des electrons positifs et negatifs. J. Phys. Radium 7, 347–353 (1936) https://doi.org/10.1051/jphysrad:0193600708034700 Born and Infeld [1934] Born, M., Infeld, L.: Foundations of the new field theory. Proc. Roy. Soc. Lond. A 144(852), 425–451 (1934) https://doi.org/10.1098/rspa.1934.0059 Dirac [1962] Dirac, P.A.M.: An Extensible model of the electron. Proc. Roy. Soc. Lond. A 268, 57–67 (1962) https://doi.org/10.1098/rspa.1962.0124 Tasinato [2014] Tasinato, G.: Cosmic Acceleration from Abelian Symmetry Breaking. JHEP 04, 067 (2014) https://doi.org/10.1007/JHEP04(2014)067 arXiv:1402.6450 [hep-th] Heisenberg [2014] Heisenberg, L.: Generalization of the Proca Action. JCAP 05, 015 (2014) https://doi.org/10.1088/1475-7516/2014/05/015 arXiv:1402.7026 [hep-th] Hull et al. [2016] Hull, M., Koyama, K., Tasinato, G.: Covariantized vector Galileons. Phys. Rev. D 93(6), 064012 (2016) https://doi.org/10.1103/PhysRevD.93.064012 arXiv:1510.07029 [hep-th] Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized Proca action for an Abelian vector field. JCAP 02, 004 (2016) https://doi.org/10.1088/1475-7516/2016/02/004 arXiv:1511.03101 [hep-th] Beltran Jimenez and Heisenberg [2016] Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodríguez, Y., Navarro, A.A.: Non-Abelian S𝑆Sitalic_S-term dark energy and inflation. Phys. Dark Univ. 19, 129–136 (2018) https://doi.org/10.1016/j.dark.2018.01.003 arXiv:1711.01935 [gr-qc] Gómez and Rodríguez [2019] Gómez, L.G., Rodríguez, Y.: Stability Conditions in the Generalized SU(2) Proca Theory. Phys. Rev. D 100(8), 084048 (2019) https://doi.org/10.1103/PhysRevD.100.084048 arXiv:1907.07961 [gr-qc] Garnica et al. [2022] Garnica, J.C., Gomez, L.G., Navarro, A.A., Rodriguez, Y.: Constant-Roll Inflation in the Generalized SU(2) Proca Theory. Annalen Phys. 534(2), 2100453 (2022) https://doi.org/10.1002/andp.202100453 arXiv:2109.10154 [gr-qc] Proca [1936] Proca, A.: Sur la theorie ondulatoire des electrons positifs et negatifs. J. Phys. Radium 7, 347–353 (1936) https://doi.org/10.1051/jphysrad:0193600708034700 Born and Infeld [1934] Born, M., Infeld, L.: Foundations of the new field theory. Proc. Roy. Soc. Lond. A 144(852), 425–451 (1934) https://doi.org/10.1098/rspa.1934.0059 Dirac [1962] Dirac, P.A.M.: An Extensible model of the electron. Proc. Roy. Soc. Lond. A 268, 57–67 (1962) https://doi.org/10.1098/rspa.1962.0124 Tasinato [2014] Tasinato, G.: Cosmic Acceleration from Abelian Symmetry Breaking. JHEP 04, 067 (2014) https://doi.org/10.1007/JHEP04(2014)067 arXiv:1402.6450 [hep-th] Heisenberg [2014] Heisenberg, L.: Generalization of the Proca Action. JCAP 05, 015 (2014) https://doi.org/10.1088/1475-7516/2014/05/015 arXiv:1402.7026 [hep-th] Hull et al. [2016] Hull, M., Koyama, K., Tasinato, G.: Covariantized vector Galileons. Phys. Rev. D 93(6), 064012 (2016) https://doi.org/10.1103/PhysRevD.93.064012 arXiv:1510.07029 [hep-th] Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized Proca action for an Abelian vector field. JCAP 02, 004 (2016) https://doi.org/10.1088/1475-7516/2016/02/004 arXiv:1511.03101 [hep-th] Beltran Jimenez and Heisenberg [2016] Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. 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(2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Garnica, J.C., Gomez, L.G., Navarro, A.A., Rodriguez, Y.: Constant-Roll Inflation in the Generalized SU(2) Proca Theory. Annalen Phys. 534(2), 2100453 (2022) https://doi.org/10.1002/andp.202100453 arXiv:2109.10154 [gr-qc] Proca [1936] Proca, A.: Sur la theorie ondulatoire des electrons positifs et negatifs. J. Phys. Radium 7, 347–353 (1936) https://doi.org/10.1051/jphysrad:0193600708034700 Born and Infeld [1934] Born, M., Infeld, L.: Foundations of the new field theory. Proc. Roy. Soc. Lond. A 144(852), 425–451 (1934) https://doi.org/10.1098/rspa.1934.0059 Dirac [1962] Dirac, P.A.M.: An Extensible model of the electron. Proc. Roy. Soc. Lond. A 268, 57–67 (1962) https://doi.org/10.1098/rspa.1962.0124 Tasinato [2014] Tasinato, G.: Cosmic Acceleration from Abelian Symmetry Breaking. JHEP 04, 067 (2014) https://doi.org/10.1007/JHEP04(2014)067 arXiv:1402.6450 [hep-th] Heisenberg [2014] Heisenberg, L.: Generalization of the Proca Action. JCAP 05, 015 (2014) https://doi.org/10.1088/1475-7516/2014/05/015 arXiv:1402.7026 [hep-th] Hull et al. [2016] Hull, M., Koyama, K., Tasinato, G.: Covariantized vector Galileons. Phys. Rev. D 93(6), 064012 (2016) https://doi.org/10.1103/PhysRevD.93.064012 arXiv:1510.07029 [hep-th] Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized Proca action for an Abelian vector field. JCAP 02, 004 (2016) https://doi.org/10.1088/1475-7516/2016/02/004 arXiv:1511.03101 [hep-th] Beltran Jimenez and Heisenberg [2016] Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. 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Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. 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Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. 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Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. 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In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. 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Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. 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Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. 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Dark Univ. 19, 129–136 (2018) https://doi.org/10.1016/j.dark.2018.01.003 arXiv:1711.01935 [gr-qc] Gómez and Rodríguez [2019] Gómez, L.G., Rodríguez, Y.: Stability Conditions in the Generalized SU(2) Proca Theory. Phys. Rev. D 100(8), 084048 (2019) https://doi.org/10.1103/PhysRevD.100.084048 arXiv:1907.07961 [gr-qc] Garnica et al. [2022] Garnica, J.C., Gomez, L.G., Navarro, A.A., Rodriguez, Y.: Constant-Roll Inflation in the Generalized SU(2) Proca Theory. Annalen Phys. 534(2), 2100453 (2022) https://doi.org/10.1002/andp.202100453 arXiv:2109.10154 [gr-qc] Proca [1936] Proca, A.: Sur la theorie ondulatoire des electrons positifs et negatifs. J. Phys. Radium 7, 347–353 (1936) https://doi.org/10.1051/jphysrad:0193600708034700 Born and Infeld [1934] Born, M., Infeld, L.: Foundations of the new field theory. Proc. Roy. Soc. Lond. A 144(852), 425–451 (1934) https://doi.org/10.1098/rspa.1934.0059 Dirac [1962] Dirac, P.A.M.: An Extensible model of the electron. Proc. Roy. Soc. Lond. A 268, 57–67 (1962) https://doi.org/10.1098/rspa.1962.0124 Tasinato [2014] Tasinato, G.: Cosmic Acceleration from Abelian Symmetry Breaking. JHEP 04, 067 (2014) https://doi.org/10.1007/JHEP04(2014)067 arXiv:1402.6450 [hep-th] Heisenberg [2014] Heisenberg, L.: Generalization of the Proca Action. JCAP 05, 015 (2014) https://doi.org/10.1088/1475-7516/2014/05/015 arXiv:1402.7026 [hep-th] Hull et al. [2016] Hull, M., Koyama, K., Tasinato, G.: Covariantized vector Galileons. Phys. Rev. D 93(6), 064012 (2016) https://doi.org/10.1103/PhysRevD.93.064012 arXiv:1510.07029 [hep-th] Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized Proca action for an Abelian vector field. JCAP 02, 004 (2016) https://doi.org/10.1088/1475-7516/2016/02/004 arXiv:1511.03101 [hep-th] Beltran Jimenez and Heisenberg [2016] Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodríguez, Y., Navarro, A.A.: Non-Abelian S𝑆Sitalic_S-term dark energy and inflation. Phys. Dark Univ. 19, 129–136 (2018) https://doi.org/10.1016/j.dark.2018.01.003 arXiv:1711.01935 [gr-qc] Gómez and Rodríguez [2019] Gómez, L.G., Rodríguez, Y.: Stability Conditions in the Generalized SU(2) Proca Theory. Phys. Rev. D 100(8), 084048 (2019) https://doi.org/10.1103/PhysRevD.100.084048 arXiv:1907.07961 [gr-qc] Garnica et al. [2022] Garnica, J.C., Gomez, L.G., Navarro, A.A., Rodriguez, Y.: Constant-Roll Inflation in the Generalized SU(2) Proca Theory. Annalen Phys. 534(2), 2100453 (2022) https://doi.org/10.1002/andp.202100453 arXiv:2109.10154 [gr-qc] Proca [1936] Proca, A.: Sur la theorie ondulatoire des electrons positifs et negatifs. J. Phys. Radium 7, 347–353 (1936) https://doi.org/10.1051/jphysrad:0193600708034700 Born and Infeld [1934] Born, M., Infeld, L.: Foundations of the new field theory. Proc. Roy. Soc. Lond. A 144(852), 425–451 (1934) https://doi.org/10.1098/rspa.1934.0059 Dirac [1962] Dirac, P.A.M.: An Extensible model of the electron. Proc. Roy. Soc. Lond. A 268, 57–67 (1962) https://doi.org/10.1098/rspa.1962.0124 Tasinato [2014] Tasinato, G.: Cosmic Acceleration from Abelian Symmetry Breaking. JHEP 04, 067 (2014) https://doi.org/10.1007/JHEP04(2014)067 arXiv:1402.6450 [hep-th] Heisenberg [2014] Heisenberg, L.: Generalization of the Proca Action. JCAP 05, 015 (2014) https://doi.org/10.1088/1475-7516/2014/05/015 arXiv:1402.7026 [hep-th] Hull et al. [2016] Hull, M., Koyama, K., Tasinato, G.: Covariantized vector Galileons. Phys. Rev. D 93(6), 064012 (2016) https://doi.org/10.1103/PhysRevD.93.064012 arXiv:1510.07029 [hep-th] Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized Proca action for an Abelian vector field. JCAP 02, 004 (2016) https://doi.org/10.1088/1475-7516/2016/02/004 arXiv:1511.03101 [hep-th] Beltran Jimenez and Heisenberg [2016] Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. 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(2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. 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D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Garnica, J.C., Gomez, L.G., Navarro, A.A., Rodriguez, Y.: Constant-Roll Inflation in the Generalized SU(2) Proca Theory. Annalen Phys. 534(2), 2100453 (2022) https://doi.org/10.1002/andp.202100453 arXiv:2109.10154 [gr-qc] Proca [1936] Proca, A.: Sur la theorie ondulatoire des electrons positifs et negatifs. J. Phys. Radium 7, 347–353 (1936) https://doi.org/10.1051/jphysrad:0193600708034700 Born and Infeld [1934] Born, M., Infeld, L.: Foundations of the new field theory. Proc. Roy. Soc. Lond. A 144(852), 425–451 (1934) https://doi.org/10.1098/rspa.1934.0059 Dirac [1962] Dirac, P.A.M.: An Extensible model of the electron. Proc. Roy. Soc. Lond. A 268, 57–67 (1962) https://doi.org/10.1098/rspa.1962.0124 Tasinato [2014] Tasinato, G.: Cosmic Acceleration from Abelian Symmetry Breaking. JHEP 04, 067 (2014) https://doi.org/10.1007/JHEP04(2014)067 arXiv:1402.6450 [hep-th] Heisenberg [2014] Heisenberg, L.: Generalization of the Proca Action. JCAP 05, 015 (2014) https://doi.org/10.1088/1475-7516/2014/05/015 arXiv:1402.7026 [hep-th] Hull et al. [2016] Hull, M., Koyama, K., Tasinato, G.: Covariantized vector Galileons. Phys. Rev. D 93(6), 064012 (2016) https://doi.org/10.1103/PhysRevD.93.064012 arXiv:1510.07029 [hep-th] Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized Proca action for an Abelian vector field. JCAP 02, 004 (2016) https://doi.org/10.1088/1475-7516/2016/02/004 arXiv:1511.03101 [hep-th] Beltran Jimenez and Heisenberg [2016] Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Proca, A.: Sur la theorie ondulatoire des electrons positifs et negatifs. J. Phys. Radium 7, 347–353 (1936) https://doi.org/10.1051/jphysrad:0193600708034700 Born and Infeld [1934] Born, M., Infeld, L.: Foundations of the new field theory. Proc. Roy. Soc. Lond. A 144(852), 425–451 (1934) https://doi.org/10.1098/rspa.1934.0059 Dirac [1962] Dirac, P.A.M.: An Extensible model of the electron. Proc. Roy. Soc. Lond. A 268, 57–67 (1962) https://doi.org/10.1098/rspa.1962.0124 Tasinato [2014] Tasinato, G.: Cosmic Acceleration from Abelian Symmetry Breaking. JHEP 04, 067 (2014) https://doi.org/10.1007/JHEP04(2014)067 arXiv:1402.6450 [hep-th] Heisenberg [2014] Heisenberg, L.: Generalization of the Proca Action. JCAP 05, 015 (2014) https://doi.org/10.1088/1475-7516/2014/05/015 arXiv:1402.7026 [hep-th] Hull et al. [2016] Hull, M., Koyama, K., Tasinato, G.: Covariantized vector Galileons. Phys. Rev. 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Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. 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B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. 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Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. 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B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. 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[2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. 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Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. 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[2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. 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Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. 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[2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. 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JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Perlmutter, S., et al.: Measurements of ΩΩ\Omegaroman_Ω and ΛΛ\Lambdaroman_Λ from 42 high redshift supernovae. Astrophys. J. 517, 565–586 (1999) https://doi.org/10.1086/307221 arXiv:astro-ph/9812133 Riess et al. [1998] Riess, A.G., et al.: Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J. 116, 1009–1038 (1998) https://doi.org/10.1086/300499 arXiv:astro-ph/9805201 Rodríguez and Navarro [2018] Rodríguez, Y., Navarro, A.A.: Non-Abelian S𝑆Sitalic_S-term dark energy and inflation. Phys. Dark Univ. 19, 129–136 (2018) https://doi.org/10.1016/j.dark.2018.01.003 arXiv:1711.01935 [gr-qc] Gómez and Rodríguez [2019] Gómez, L.G., Rodríguez, Y.: Stability Conditions in the Generalized SU(2) Proca Theory. Phys. Rev. D 100(8), 084048 (2019) https://doi.org/10.1103/PhysRevD.100.084048 arXiv:1907.07961 [gr-qc] Garnica et al. [2022] Garnica, J.C., Gomez, L.G., Navarro, A.A., Rodriguez, Y.: Constant-Roll Inflation in the Generalized SU(2) Proca Theory. Annalen Phys. 534(2), 2100453 (2022) https://doi.org/10.1002/andp.202100453 arXiv:2109.10154 [gr-qc] Proca [1936] Proca, A.: Sur la theorie ondulatoire des electrons positifs et negatifs. J. Phys. Radium 7, 347–353 (1936) https://doi.org/10.1051/jphysrad:0193600708034700 Born and Infeld [1934] Born, M., Infeld, L.: Foundations of the new field theory. Proc. Roy. Soc. Lond. A 144(852), 425–451 (1934) https://doi.org/10.1098/rspa.1934.0059 Dirac [1962] Dirac, P.A.M.: An Extensible model of the electron. Proc. Roy. Soc. Lond. A 268, 57–67 (1962) https://doi.org/10.1098/rspa.1962.0124 Tasinato [2014] Tasinato, G.: Cosmic Acceleration from Abelian Symmetry Breaking. JHEP 04, 067 (2014) https://doi.org/10.1007/JHEP04(2014)067 arXiv:1402.6450 [hep-th] Heisenberg [2014] Heisenberg, L.: Generalization of the Proca Action. JCAP 05, 015 (2014) https://doi.org/10.1088/1475-7516/2014/05/015 arXiv:1402.7026 [hep-th] Hull et al. [2016] Hull, M., Koyama, K., Tasinato, G.: Covariantized vector Galileons. Phys. Rev. D 93(6), 064012 (2016) https://doi.org/10.1103/PhysRevD.93.064012 arXiv:1510.07029 [hep-th] Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized Proca action for an Abelian vector field. JCAP 02, 004 (2016) https://doi.org/10.1088/1475-7516/2016/02/004 arXiv:1511.03101 [hep-th] Beltran Jimenez and Heisenberg [2016] Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Riess, A.G., et al.: Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J. 116, 1009–1038 (1998) https://doi.org/10.1086/300499 arXiv:astro-ph/9805201 Rodríguez and Navarro [2018] Rodríguez, Y., Navarro, A.A.: Non-Abelian S𝑆Sitalic_S-term dark energy and inflation. Phys. Dark Univ. 19, 129–136 (2018) https://doi.org/10.1016/j.dark.2018.01.003 arXiv:1711.01935 [gr-qc] Gómez and Rodríguez [2019] Gómez, L.G., Rodríguez, Y.: Stability Conditions in the Generalized SU(2) Proca Theory. Phys. Rev. D 100(8), 084048 (2019) https://doi.org/10.1103/PhysRevD.100.084048 arXiv:1907.07961 [gr-qc] Garnica et al. [2022] Garnica, J.C., Gomez, L.G., Navarro, A.A., Rodriguez, Y.: Constant-Roll Inflation in the Generalized SU(2) Proca Theory. Annalen Phys. 534(2), 2100453 (2022) https://doi.org/10.1002/andp.202100453 arXiv:2109.10154 [gr-qc] Proca [1936] Proca, A.: Sur la theorie ondulatoire des electrons positifs et negatifs. J. Phys. Radium 7, 347–353 (1936) https://doi.org/10.1051/jphysrad:0193600708034700 Born and Infeld [1934] Born, M., Infeld, L.: Foundations of the new field theory. Proc. Roy. Soc. Lond. A 144(852), 425–451 (1934) https://doi.org/10.1098/rspa.1934.0059 Dirac [1962] Dirac, P.A.M.: An Extensible model of the electron. Proc. Roy. Soc. Lond. A 268, 57–67 (1962) https://doi.org/10.1098/rspa.1962.0124 Tasinato [2014] Tasinato, G.: Cosmic Acceleration from Abelian Symmetry Breaking. JHEP 04, 067 (2014) https://doi.org/10.1007/JHEP04(2014)067 arXiv:1402.6450 [hep-th] Heisenberg [2014] Heisenberg, L.: Generalization of the Proca Action. JCAP 05, 015 (2014) https://doi.org/10.1088/1475-7516/2014/05/015 arXiv:1402.7026 [hep-th] Hull et al. [2016] Hull, M., Koyama, K., Tasinato, G.: Covariantized vector Galileons. Phys. Rev. D 93(6), 064012 (2016) https://doi.org/10.1103/PhysRevD.93.064012 arXiv:1510.07029 [hep-th] Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized Proca action for an Abelian vector field. JCAP 02, 004 (2016) https://doi.org/10.1088/1475-7516/2016/02/004 arXiv:1511.03101 [hep-th] Beltran Jimenez and Heisenberg [2016] Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. 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JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. 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Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. 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A 268, 57–67 (1962) https://doi.org/10.1098/rspa.1962.0124 Tasinato [2014] Tasinato, G.: Cosmic Acceleration from Abelian Symmetry Breaking. JHEP 04, 067 (2014) https://doi.org/10.1007/JHEP04(2014)067 arXiv:1402.6450 [hep-th] Heisenberg [2014] Heisenberg, L.: Generalization of the Proca Action. JCAP 05, 015 (2014) https://doi.org/10.1088/1475-7516/2014/05/015 arXiv:1402.7026 [hep-th] Hull et al. [2016] Hull, M., Koyama, K., Tasinato, G.: Covariantized vector Galileons. Phys. Rev. D 93(6), 064012 (2016) https://doi.org/10.1103/PhysRevD.93.064012 arXiv:1510.07029 [hep-th] Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized Proca action for an Abelian vector field. JCAP 02, 004 (2016) https://doi.org/10.1088/1475-7516/2016/02/004 arXiv:1511.03101 [hep-th] Beltran Jimenez and Heisenberg [2016] Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. 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B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. 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B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. 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D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. 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B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. 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B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. 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Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. 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In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. 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Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. 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[2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. 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D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Riess, A.G., et al.: Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J. 116, 1009–1038 (1998) https://doi.org/10.1086/300499 arXiv:astro-ph/9805201 Rodríguez and Navarro [2018] Rodríguez, Y., Navarro, A.A.: Non-Abelian S𝑆Sitalic_S-term dark energy and inflation. Phys. 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A 268, 57–67 (1962) https://doi.org/10.1098/rspa.1962.0124 Tasinato [2014] Tasinato, G.: Cosmic Acceleration from Abelian Symmetry Breaking. JHEP 04, 067 (2014) https://doi.org/10.1007/JHEP04(2014)067 arXiv:1402.6450 [hep-th] Heisenberg [2014] Heisenberg, L.: Generalization of the Proca Action. JCAP 05, 015 (2014) https://doi.org/10.1088/1475-7516/2014/05/015 arXiv:1402.7026 [hep-th] Hull et al. [2016] Hull, M., Koyama, K., Tasinato, G.: Covariantized vector Galileons. Phys. Rev. D 93(6), 064012 (2016) https://doi.org/10.1103/PhysRevD.93.064012 arXiv:1510.07029 [hep-th] Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized Proca action for an Abelian vector field. JCAP 02, 004 (2016) https://doi.org/10.1088/1475-7516/2016/02/004 arXiv:1511.03101 [hep-th] Beltran Jimenez and Heisenberg [2016] Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. 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JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. 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Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. 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A 268, 57–67 (1962) https://doi.org/10.1098/rspa.1962.0124 Tasinato [2014] Tasinato, G.: Cosmic Acceleration from Abelian Symmetry Breaking. JHEP 04, 067 (2014) https://doi.org/10.1007/JHEP04(2014)067 arXiv:1402.6450 [hep-th] Heisenberg [2014] Heisenberg, L.: Generalization of the Proca Action. JCAP 05, 015 (2014) https://doi.org/10.1088/1475-7516/2014/05/015 arXiv:1402.7026 [hep-th] Hull et al. [2016] Hull, M., Koyama, K., Tasinato, G.: Covariantized vector Galileons. Phys. Rev. D 93(6), 064012 (2016) https://doi.org/10.1103/PhysRevD.93.064012 arXiv:1510.07029 [hep-th] Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized Proca action for an Abelian vector field. JCAP 02, 004 (2016) https://doi.org/10.1088/1475-7516/2016/02/004 arXiv:1511.03101 [hep-th] Beltran Jimenez and Heisenberg [2016] Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. 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B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. 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B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. 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D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. 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B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. 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B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. 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Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. 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In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. 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Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. 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[2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. 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D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. 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Dark Univ. 19, 129–136 (2018) https://doi.org/10.1016/j.dark.2018.01.003 arXiv:1711.01935 [gr-qc] Gómez and Rodríguez [2019] Gómez, L.G., Rodríguez, Y.: Stability Conditions in the Generalized SU(2) Proca Theory. Phys. Rev. D 100(8), 084048 (2019) https://doi.org/10.1103/PhysRevD.100.084048 arXiv:1907.07961 [gr-qc] Garnica et al. [2022] Garnica, J.C., Gomez, L.G., Navarro, A.A., Rodriguez, Y.: Constant-Roll Inflation in the Generalized SU(2) Proca Theory. Annalen Phys. 534(2), 2100453 (2022) https://doi.org/10.1002/andp.202100453 arXiv:2109.10154 [gr-qc] Proca [1936] Proca, A.: Sur la theorie ondulatoire des electrons positifs et negatifs. J. Phys. Radium 7, 347–353 (1936) https://doi.org/10.1051/jphysrad:0193600708034700 Born and Infeld [1934] Born, M., Infeld, L.: Foundations of the new field theory. Proc. Roy. Soc. Lond. A 144(852), 425–451 (1934) https://doi.org/10.1098/rspa.1934.0059 Dirac [1962] Dirac, P.A.M.: An Extensible model of the electron. Proc. Roy. Soc. Lond. A 268, 57–67 (1962) https://doi.org/10.1098/rspa.1962.0124 Tasinato [2014] Tasinato, G.: Cosmic Acceleration from Abelian Symmetry Breaking. JHEP 04, 067 (2014) https://doi.org/10.1007/JHEP04(2014)067 arXiv:1402.6450 [hep-th] Heisenberg [2014] Heisenberg, L.: Generalization of the Proca Action. JCAP 05, 015 (2014) https://doi.org/10.1088/1475-7516/2014/05/015 arXiv:1402.7026 [hep-th] Hull et al. [2016] Hull, M., Koyama, K., Tasinato, G.: Covariantized vector Galileons. Phys. Rev. D 93(6), 064012 (2016) https://doi.org/10.1103/PhysRevD.93.064012 arXiv:1510.07029 [hep-th] Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized Proca action for an Abelian vector field. JCAP 02, 004 (2016) https://doi.org/10.1088/1475-7516/2016/02/004 arXiv:1511.03101 [hep-th] Beltran Jimenez and Heisenberg [2016] Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodríguez, Y., Navarro, A.A.: Non-Abelian S𝑆Sitalic_S-term dark energy and inflation. Phys. Dark Univ. 19, 129–136 (2018) https://doi.org/10.1016/j.dark.2018.01.003 arXiv:1711.01935 [gr-qc] Gómez and Rodríguez [2019] Gómez, L.G., Rodríguez, Y.: Stability Conditions in the Generalized SU(2) Proca Theory. Phys. Rev. D 100(8), 084048 (2019) https://doi.org/10.1103/PhysRevD.100.084048 arXiv:1907.07961 [gr-qc] Garnica et al. [2022] Garnica, J.C., Gomez, L.G., Navarro, A.A., Rodriguez, Y.: Constant-Roll Inflation in the Generalized SU(2) Proca Theory. Annalen Phys. 534(2), 2100453 (2022) https://doi.org/10.1002/andp.202100453 arXiv:2109.10154 [gr-qc] Proca [1936] Proca, A.: Sur la theorie ondulatoire des electrons positifs et negatifs. J. Phys. Radium 7, 347–353 (1936) https://doi.org/10.1051/jphysrad:0193600708034700 Born and Infeld [1934] Born, M., Infeld, L.: Foundations of the new field theory. Proc. Roy. Soc. Lond. A 144(852), 425–451 (1934) https://doi.org/10.1098/rspa.1934.0059 Dirac [1962] Dirac, P.A.M.: An Extensible model of the electron. Proc. Roy. Soc. Lond. A 268, 57–67 (1962) https://doi.org/10.1098/rspa.1962.0124 Tasinato [2014] Tasinato, G.: Cosmic Acceleration from Abelian Symmetry Breaking. JHEP 04, 067 (2014) https://doi.org/10.1007/JHEP04(2014)067 arXiv:1402.6450 [hep-th] Heisenberg [2014] Heisenberg, L.: Generalization of the Proca Action. JCAP 05, 015 (2014) https://doi.org/10.1088/1475-7516/2014/05/015 arXiv:1402.7026 [hep-th] Hull et al. [2016] Hull, M., Koyama, K., Tasinato, G.: Covariantized vector Galileons. Phys. Rev. D 93(6), 064012 (2016) https://doi.org/10.1103/PhysRevD.93.064012 arXiv:1510.07029 [hep-th] Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized Proca action for an Abelian vector field. JCAP 02, 004 (2016) https://doi.org/10.1088/1475-7516/2016/02/004 arXiv:1511.03101 [hep-th] Beltran Jimenez and Heisenberg [2016] Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. 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(2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. 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D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Garnica, J.C., Gomez, L.G., Navarro, A.A., Rodriguez, Y.: Constant-Roll Inflation in the Generalized SU(2) Proca Theory. Annalen Phys. 534(2), 2100453 (2022) https://doi.org/10.1002/andp.202100453 arXiv:2109.10154 [gr-qc] Proca [1936] Proca, A.: Sur la theorie ondulatoire des electrons positifs et negatifs. J. Phys. Radium 7, 347–353 (1936) https://doi.org/10.1051/jphysrad:0193600708034700 Born and Infeld [1934] Born, M., Infeld, L.: Foundations of the new field theory. Proc. Roy. Soc. Lond. A 144(852), 425–451 (1934) https://doi.org/10.1098/rspa.1934.0059 Dirac [1962] Dirac, P.A.M.: An Extensible model of the electron. Proc. Roy. Soc. Lond. A 268, 57–67 (1962) https://doi.org/10.1098/rspa.1962.0124 Tasinato [2014] Tasinato, G.: Cosmic Acceleration from Abelian Symmetry Breaking. JHEP 04, 067 (2014) https://doi.org/10.1007/JHEP04(2014)067 arXiv:1402.6450 [hep-th] Heisenberg [2014] Heisenberg, L.: Generalization of the Proca Action. JCAP 05, 015 (2014) https://doi.org/10.1088/1475-7516/2014/05/015 arXiv:1402.7026 [hep-th] Hull et al. [2016] Hull, M., Koyama, K., Tasinato, G.: Covariantized vector Galileons. Phys. Rev. D 93(6), 064012 (2016) https://doi.org/10.1103/PhysRevD.93.064012 arXiv:1510.07029 [hep-th] Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized Proca action for an Abelian vector field. JCAP 02, 004 (2016) https://doi.org/10.1088/1475-7516/2016/02/004 arXiv:1511.03101 [hep-th] Beltran Jimenez and Heisenberg [2016] Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Proca, A.: Sur la theorie ondulatoire des electrons positifs et negatifs. J. Phys. Radium 7, 347–353 (1936) https://doi.org/10.1051/jphysrad:0193600708034700 Born and Infeld [1934] Born, M., Infeld, L.: Foundations of the new field theory. Proc. Roy. Soc. Lond. A 144(852), 425–451 (1934) https://doi.org/10.1098/rspa.1934.0059 Dirac [1962] Dirac, P.A.M.: An Extensible model of the electron. Proc. Roy. Soc. Lond. A 268, 57–67 (1962) https://doi.org/10.1098/rspa.1962.0124 Tasinato [2014] Tasinato, G.: Cosmic Acceleration from Abelian Symmetry Breaking. JHEP 04, 067 (2014) https://doi.org/10.1007/JHEP04(2014)067 arXiv:1402.6450 [hep-th] Heisenberg [2014] Heisenberg, L.: Generalization of the Proca Action. JCAP 05, 015 (2014) https://doi.org/10.1088/1475-7516/2014/05/015 arXiv:1402.7026 [hep-th] Hull et al. [2016] Hull, M., Koyama, K., Tasinato, G.: Covariantized vector Galileons. Phys. Rev. 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Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. 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B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. 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Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. 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B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. 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[2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. 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Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. 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[2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. 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Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. 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Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Riess, A.G., et al.: Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J. 116, 1009–1038 (1998) https://doi.org/10.1086/300499 arXiv:astro-ph/9805201 Rodríguez and Navarro [2018] Rodríguez, Y., Navarro, A.A.: Non-Abelian S𝑆Sitalic_S-term dark energy and inflation. Phys. Dark Univ. 19, 129–136 (2018) https://doi.org/10.1016/j.dark.2018.01.003 arXiv:1711.01935 [gr-qc] Gómez and Rodríguez [2019] Gómez, L.G., Rodríguez, Y.: Stability Conditions in the Generalized SU(2) Proca Theory. Phys. Rev. D 100(8), 084048 (2019) https://doi.org/10.1103/PhysRevD.100.084048 arXiv:1907.07961 [gr-qc] Garnica et al. [2022] Garnica, J.C., Gomez, L.G., Navarro, A.A., Rodriguez, Y.: Constant-Roll Inflation in the Generalized SU(2) Proca Theory. Annalen Phys. 534(2), 2100453 (2022) https://doi.org/10.1002/andp.202100453 arXiv:2109.10154 [gr-qc] Proca [1936] Proca, A.: Sur la theorie ondulatoire des electrons positifs et negatifs. J. Phys. Radium 7, 347–353 (1936) https://doi.org/10.1051/jphysrad:0193600708034700 Born and Infeld [1934] Born, M., Infeld, L.: Foundations of the new field theory. Proc. Roy. Soc. Lond. A 144(852), 425–451 (1934) https://doi.org/10.1098/rspa.1934.0059 Dirac [1962] Dirac, P.A.M.: An Extensible model of the electron. Proc. Roy. Soc. Lond. A 268, 57–67 (1962) https://doi.org/10.1098/rspa.1962.0124 Tasinato [2014] Tasinato, G.: Cosmic Acceleration from Abelian Symmetry Breaking. JHEP 04, 067 (2014) https://doi.org/10.1007/JHEP04(2014)067 arXiv:1402.6450 [hep-th] Heisenberg [2014] Heisenberg, L.: Generalization of the Proca Action. JCAP 05, 015 (2014) https://doi.org/10.1088/1475-7516/2014/05/015 arXiv:1402.7026 [hep-th] Hull et al. [2016] Hull, M., Koyama, K., Tasinato, G.: Covariantized vector Galileons. Phys. Rev. D 93(6), 064012 (2016) https://doi.org/10.1103/PhysRevD.93.064012 arXiv:1510.07029 [hep-th] Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized Proca action for an Abelian vector field. JCAP 02, 004 (2016) https://doi.org/10.1088/1475-7516/2016/02/004 arXiv:1511.03101 [hep-th] Beltran Jimenez and Heisenberg [2016] Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodríguez, Y., Navarro, A.A.: Non-Abelian S𝑆Sitalic_S-term dark energy and inflation. Phys. Dark Univ. 19, 129–136 (2018) https://doi.org/10.1016/j.dark.2018.01.003 arXiv:1711.01935 [gr-qc] Gómez and Rodríguez [2019] Gómez, L.G., Rodríguez, Y.: Stability Conditions in the Generalized SU(2) Proca Theory. Phys. Rev. D 100(8), 084048 (2019) https://doi.org/10.1103/PhysRevD.100.084048 arXiv:1907.07961 [gr-qc] Garnica et al. [2022] Garnica, J.C., Gomez, L.G., Navarro, A.A., Rodriguez, Y.: Constant-Roll Inflation in the Generalized SU(2) Proca Theory. Annalen Phys. 534(2), 2100453 (2022) https://doi.org/10.1002/andp.202100453 arXiv:2109.10154 [gr-qc] Proca [1936] Proca, A.: Sur la theorie ondulatoire des electrons positifs et negatifs. J. Phys. Radium 7, 347–353 (1936) https://doi.org/10.1051/jphysrad:0193600708034700 Born and Infeld [1934] Born, M., Infeld, L.: Foundations of the new field theory. Proc. Roy. Soc. Lond. A 144(852), 425–451 (1934) https://doi.org/10.1098/rspa.1934.0059 Dirac [1962] Dirac, P.A.M.: An Extensible model of the electron. Proc. Roy. Soc. Lond. A 268, 57–67 (1962) https://doi.org/10.1098/rspa.1962.0124 Tasinato [2014] Tasinato, G.: Cosmic Acceleration from Abelian Symmetry Breaking. JHEP 04, 067 (2014) https://doi.org/10.1007/JHEP04(2014)067 arXiv:1402.6450 [hep-th] Heisenberg [2014] Heisenberg, L.: Generalization of the Proca Action. JCAP 05, 015 (2014) https://doi.org/10.1088/1475-7516/2014/05/015 arXiv:1402.7026 [hep-th] Hull et al. [2016] Hull, M., Koyama, K., Tasinato, G.: Covariantized vector Galileons. Phys. Rev. D 93(6), 064012 (2016) https://doi.org/10.1103/PhysRevD.93.064012 arXiv:1510.07029 [hep-th] Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized Proca action for an Abelian vector field. 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(2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Garnica, J.C., Gomez, L.G., Navarro, A.A., Rodriguez, Y.: Constant-Roll Inflation in the Generalized SU(2) Proca Theory. Annalen Phys. 534(2), 2100453 (2022) https://doi.org/10.1002/andp.202100453 arXiv:2109.10154 [gr-qc] Proca [1936] Proca, A.: Sur la theorie ondulatoire des electrons positifs et negatifs. J. Phys. Radium 7, 347–353 (1936) https://doi.org/10.1051/jphysrad:0193600708034700 Born and Infeld [1934] Born, M., Infeld, L.: Foundations of the new field theory. Proc. Roy. Soc. Lond. A 144(852), 425–451 (1934) https://doi.org/10.1098/rspa.1934.0059 Dirac [1962] Dirac, P.A.M.: An Extensible model of the electron. Proc. Roy. Soc. Lond. A 268, 57–67 (1962) https://doi.org/10.1098/rspa.1962.0124 Tasinato [2014] Tasinato, G.: Cosmic Acceleration from Abelian Symmetry Breaking. JHEP 04, 067 (2014) https://doi.org/10.1007/JHEP04(2014)067 arXiv:1402.6450 [hep-th] Heisenberg [2014] Heisenberg, L.: Generalization of the Proca Action. JCAP 05, 015 (2014) https://doi.org/10.1088/1475-7516/2014/05/015 arXiv:1402.7026 [hep-th] Hull et al. [2016] Hull, M., Koyama, K., Tasinato, G.: Covariantized vector Galileons. Phys. Rev. D 93(6), 064012 (2016) https://doi.org/10.1103/PhysRevD.93.064012 arXiv:1510.07029 [hep-th] Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized Proca action for an Abelian vector field. JCAP 02, 004 (2016) https://doi.org/10.1088/1475-7516/2016/02/004 arXiv:1511.03101 [hep-th] Beltran Jimenez and Heisenberg [2016] Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. 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Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. 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Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. 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Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. 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In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. 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Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. 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Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. 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Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. 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A 144(852), 425–451 (1934) https://doi.org/10.1098/rspa.1934.0059 Dirac [1962] Dirac, P.A.M.: An Extensible model of the electron. Proc. Roy. Soc. Lond. A 268, 57–67 (1962) https://doi.org/10.1098/rspa.1962.0124 Tasinato [2014] Tasinato, G.: Cosmic Acceleration from Abelian Symmetry Breaking. JHEP 04, 067 (2014) https://doi.org/10.1007/JHEP04(2014)067 arXiv:1402.6450 [hep-th] Heisenberg [2014] Heisenberg, L.: Generalization of the Proca Action. JCAP 05, 015 (2014) https://doi.org/10.1088/1475-7516/2014/05/015 arXiv:1402.7026 [hep-th] Hull et al. [2016] Hull, M., Koyama, K., Tasinato, G.: Covariantized vector Galileons. Phys. Rev. D 93(6), 064012 (2016) https://doi.org/10.1103/PhysRevD.93.064012 arXiv:1510.07029 [hep-th] Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized Proca action for an Abelian vector field. JCAP 02, 004 (2016) https://doi.org/10.1088/1475-7516/2016/02/004 arXiv:1511.03101 [hep-th] Beltran Jimenez and Heisenberg [2016] Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. 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(2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. 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A 268, 57–67 (1962) https://doi.org/10.1098/rspa.1962.0124 Tasinato [2014] Tasinato, G.: Cosmic Acceleration from Abelian Symmetry Breaking. JHEP 04, 067 (2014) https://doi.org/10.1007/JHEP04(2014)067 arXiv:1402.6450 [hep-th] Heisenberg [2014] Heisenberg, L.: Generalization of the Proca Action. JCAP 05, 015 (2014) https://doi.org/10.1088/1475-7516/2014/05/015 arXiv:1402.7026 [hep-th] Hull et al. [2016] Hull, M., Koyama, K., Tasinato, G.: Covariantized vector Galileons. Phys. Rev. D 93(6), 064012 (2016) https://doi.org/10.1103/PhysRevD.93.064012 arXiv:1510.07029 [hep-th] Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized Proca action for an Abelian vector field. JCAP 02, 004 (2016) https://doi.org/10.1088/1475-7516/2016/02/004 arXiv:1511.03101 [hep-th] Beltran Jimenez and Heisenberg [2016] Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. 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Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. 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D 93(6), 064012 (2016) https://doi.org/10.1103/PhysRevD.93.064012 arXiv:1510.07029 [hep-th] Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized Proca action for an Abelian vector field. JCAP 02, 004 (2016) https://doi.org/10.1088/1475-7516/2016/02/004 arXiv:1511.03101 [hep-th] Beltran Jimenez and Heisenberg [2016] Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. 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B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. 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JCAP 02, 004 (2016) https://doi.org/10.1088/1475-7516/2016/02/004 arXiv:1511.03101 [hep-th] Beltran Jimenez and Heisenberg [2016] Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. 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B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? 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Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. 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B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. 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B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. 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Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. 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D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. 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D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. 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Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. 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[2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. 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D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982)
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(2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Garnica, J.C., Gomez, L.G., Navarro, A.A., Rodriguez, Y.: Constant-Roll Inflation in the Generalized SU(2) Proca Theory. Annalen Phys. 534(2), 2100453 (2022) https://doi.org/10.1002/andp.202100453 arXiv:2109.10154 [gr-qc] Proca [1936] Proca, A.: Sur la theorie ondulatoire des electrons positifs et negatifs. J. Phys. Radium 7, 347–353 (1936) https://doi.org/10.1051/jphysrad:0193600708034700 Born and Infeld [1934] Born, M., Infeld, L.: Foundations of the new field theory. Proc. Roy. Soc. Lond. A 144(852), 425–451 (1934) https://doi.org/10.1098/rspa.1934.0059 Dirac [1962] Dirac, P.A.M.: An Extensible model of the electron. Proc. Roy. Soc. Lond. A 268, 57–67 (1962) https://doi.org/10.1098/rspa.1962.0124 Tasinato [2014] Tasinato, G.: Cosmic Acceleration from Abelian Symmetry Breaking. JHEP 04, 067 (2014) https://doi.org/10.1007/JHEP04(2014)067 arXiv:1402.6450 [hep-th] Heisenberg [2014] Heisenberg, L.: Generalization of the Proca Action. JCAP 05, 015 (2014) https://doi.org/10.1088/1475-7516/2014/05/015 arXiv:1402.7026 [hep-th] Hull et al. [2016] Hull, M., Koyama, K., Tasinato, G.: Covariantized vector Galileons. Phys. Rev. D 93(6), 064012 (2016) https://doi.org/10.1103/PhysRevD.93.064012 arXiv:1510.07029 [hep-th] Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized Proca action for an Abelian vector field. JCAP 02, 004 (2016) https://doi.org/10.1088/1475-7516/2016/02/004 arXiv:1511.03101 [hep-th] Beltran Jimenez and Heisenberg [2016] Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. 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Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. 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B 68, 22 (1982) Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. 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[2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. 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In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. 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[2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. 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Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. 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[2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. 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Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. 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A 144(852), 425–451 (1934) https://doi.org/10.1098/rspa.1934.0059 Dirac [1962] Dirac, P.A.M.: An Extensible model of the electron. Proc. Roy. Soc. Lond. A 268, 57–67 (1962) https://doi.org/10.1098/rspa.1962.0124 Tasinato [2014] Tasinato, G.: Cosmic Acceleration from Abelian Symmetry Breaking. JHEP 04, 067 (2014) https://doi.org/10.1007/JHEP04(2014)067 arXiv:1402.6450 [hep-th] Heisenberg [2014] Heisenberg, L.: Generalization of the Proca Action. JCAP 05, 015 (2014) https://doi.org/10.1088/1475-7516/2014/05/015 arXiv:1402.7026 [hep-th] Hull et al. [2016] Hull, M., Koyama, K., Tasinato, G.: Covariantized vector Galileons. Phys. Rev. D 93(6), 064012 (2016) https://doi.org/10.1103/PhysRevD.93.064012 arXiv:1510.07029 [hep-th] Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized Proca action for an Abelian vector field. JCAP 02, 004 (2016) https://doi.org/10.1088/1475-7516/2016/02/004 arXiv:1511.03101 [hep-th] Beltran Jimenez and Heisenberg [2016] Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. 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[2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. 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[2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. 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Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. 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[2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. 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B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. 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[2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. 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Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. 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Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. 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B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. 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[2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. 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[2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. 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D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. 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Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. 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[2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. 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A 268, 57–67 (1962) https://doi.org/10.1098/rspa.1962.0124 Tasinato [2014] Tasinato, G.: Cosmic Acceleration from Abelian Symmetry Breaking. JHEP 04, 067 (2014) https://doi.org/10.1007/JHEP04(2014)067 arXiv:1402.6450 [hep-th] Heisenberg [2014] Heisenberg, L.: Generalization of the Proca Action. JCAP 05, 015 (2014) https://doi.org/10.1088/1475-7516/2014/05/015 arXiv:1402.7026 [hep-th] Hull et al. [2016] Hull, M., Koyama, K., Tasinato, G.: Covariantized vector Galileons. Phys. Rev. D 93(6), 064012 (2016) https://doi.org/10.1103/PhysRevD.93.064012 arXiv:1510.07029 [hep-th] Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized Proca action for an Abelian vector field. JCAP 02, 004 (2016) https://doi.org/10.1088/1475-7516/2016/02/004 arXiv:1511.03101 [hep-th] Beltran Jimenez and Heisenberg [2016] Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. 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B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. 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B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? 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Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. 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JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. 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Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. 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D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. 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Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. 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[2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. 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Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. 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[2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. 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[2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. 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B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. 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[2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. 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B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. 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JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. 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B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. 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Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. 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In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. 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Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. 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[2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. 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D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. 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Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. 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JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? 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[2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. 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Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. 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In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. 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[2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. 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Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. 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D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. 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B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. 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B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? 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Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. 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JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. 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Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. 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D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. 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Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. 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Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. 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Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. 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B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. 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In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. 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D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. 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B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. 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[2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. 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Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. 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Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. 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B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. 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[2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. 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[2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. 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D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. 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Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. 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[2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. 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Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. 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Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. 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B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. 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In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. 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D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. 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B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? 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Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. 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Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. 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Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. 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Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Peter, P., Rodriguez, Y.: Generalized Proca action for an Abelian vector field. JCAP 02, 004 (2016) https://doi.org/10.1088/1475-7516/2016/02/004 arXiv:1511.03101 [hep-th] Beltran Jimenez and Heisenberg [2016] Beltran Jimenez, J., Heisenberg, L.: Derivative self-interactions for a massive vector field. Phys. Lett. B757, 405–411 (2016) https://doi.org/10.1016/j.physletb.2016.04.017 arXiv:1602.03410 [hep-th] Allys et al. [2016] Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. 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B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. 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B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. 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D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. 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[2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. 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Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. 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JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Beltran Almeida, J.P., Peter, P., Rodríguez, Y.: On the 4D generalized Proca action for an Abelian vector field. JCAP 09, 026 (2016) https://doi.org/10.1088/1475-7516/2016/09/026 arXiv:1605.08355 [hep-th] Rodriguez and Navarro [2017] Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? 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[2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. 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Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. 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In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. 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[2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. 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Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. 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[2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. 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D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982)
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Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? 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D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. 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[2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. 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In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. 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[2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. 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Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. 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[2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982)
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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. 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Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodriguez, Y., Navarro, A.A.: Scalar and vector Galileons. J. Phys. Conf. Ser. 831(1), 012004 (2017) https://doi.org/10.1088/1742-6596/831/1/012004 arXiv:1703.01884 [hep-th] Heisenberg [2019] Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Heisenberg, L.: A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019) https://doi.org/10.1016/j.physrep.2018.11.006 arXiv:1807.01725 [gr-qc] Sanongkhun and Vanichchapongjaroen [2020] Sanongkhun, J., Vanichchapongjaroen, P.: On constrained analysis and diffeomorphism invariance of generalised Proca theories. Gen. Rel. Grav. 52(3), 26 (2020) https://doi.org/10.1007/s10714-020-02678-y arXiv:1907.12794 [hep-th] Ostrogradsky [1850] Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. 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D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. 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D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. 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JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. 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D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. 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B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. 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B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. 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[2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. 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Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. 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Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. 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Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. 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D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. 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Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850) Heisenberg et al. [2016] Heisenberg, L., Kase, R., Tsujikawa, S.: Beyond generalized Proca theories. Phys. Lett. B 760, 617–626 (2016) https://doi.org/10.1016/j.physletb.2016.07.052 arXiv:1605.05565 [hep-th] Gallego Cadavid and Rodriguez [2019] Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. 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B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? 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[2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. 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D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. 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[2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. 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[2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. 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B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y.: A systematic procedure to build the beyond generalized Proca field theory. Phys. Lett. B 798, 134958 (2019) https://doi.org/10.1016/j.physletb.2019.134958 arXiv:1905.10664 [hep-th] Beltrán Jiménez et al. [2020] Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Beltrán Jiménez, J., Rham, C., Heisenberg, L.: Generalized Proca and its Constraint Algebra. Phys. Lett. B 802, 135244 (2020) https://doi.org/10.1016/j.physletb.2020.135244 arXiv:1906.04805 [hep-th] de Rham et al. [2022a] Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. 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In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. 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D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. 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D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. 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Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Heisenberg, L., Kumar, A., Zosso, J.: Quantum stability of a new Proca theory. Phys. Rev. D 105(2), 024033 (2022) https://doi.org/10.1103/PhysRevD.105.024033 arXiv:2108.12892 [hep-th] de Rham et al. [2022b] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V.: Cosmology of Extended Proca-Nuevo. JCAP 03, 053 (2022) https://doi.org/10.1088/1475-7516/2022/03/053 arXiv:2110.14327 [hep-th] Errasti Díez [2022] Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. 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Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. 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B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. 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D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. 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[2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. 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Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. 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Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. 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D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. 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B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. 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Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. 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[2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. 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[2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. 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D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. 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[2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. 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Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. 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D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. 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[2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. 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D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. 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[2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. 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Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V.: (Extended) Proca-Nuevo under the two-dimensional loupe (2022) arXiv:2212.02549 [hep-th] de Rham et al. [2023] Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. 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Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. 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[2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982)
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A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rham, C., Garcia-Saenz, S., Heisenberg, L., Pozsgay, V., Wang, X.: To Half–Be or Not To Be? (2023) arXiv:2303.05354 [hep-th] Errasti Díez et al. [2020a] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. 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D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Maxwell-Proca theory: Definition and construction. Phys. Rev. D 101(4), 045009 (2020) https://doi.org/10.1103/PhysRevD.101.045009 arXiv:1905.06968 [hep-th] Errasti Díez et al. [2020b] Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Gording, B., Méndez-Zavaleta, J.A., Schmidt-May, A.: Complete theory of Maxwell and Proca fields. Phys. Rev. D 101(4), 045008 (2020) https://doi.org/10.1103/PhysRevD.101.045008 arXiv:1905.06967 [hep-th] Gallego Cadavid et al. [2020] Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. 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Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. 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In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. 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Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. 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Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. 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Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. 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[2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. 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D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Gallego Cadavid, A., Rodriguez, Y., Gómez, L.G.: Generalized SU(2) Proca theory reconstructed and beyond. Phys. Rev. D 102(10), 104066 (2020) https://doi.org/10.1103/PhysRevD.102.104066 arXiv:2009.03241 [hep-th] Errasti Díez et al. [2020] Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. 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Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. 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Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. 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D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982)
- Errasti Díez, V., Maier, M., Méndez-Zavaleta, J.A., Taslimi Tehrani, M.: Lagrangian constraint analysis of first-order classical field theories with an application to gravity. Phys. Rev. D 102, 065015 (2020) https://doi.org/10.1103/PhysRevD.102.065015 arXiv:2007.11020 [hep-th] Faddeev and Jackiw [1988] Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982)
- Faddeev, L.D., Jackiw, R.: Hamiltonian Reduction of Unconstrained and Constrained Systems. Phys. Rev. Lett. 60, 1692–1694 (1988) https://doi.org/10.1103/PhysRevLett.60.1692 Jackiw [1993] Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Jackiw, R.: (Constrained) quantization without tears. In: 2nd Workshop on Constraint Theory and Quantization Methods, pp. 367–381 (1993) Barcelos-Neto and Wotzasek [1992a] Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Faddeev-Jackiw quantization and constraints. Int. J. Mod. Phys. A 7, 4981–5004 (1992) https://doi.org/10.1142/S0217751X9200226X Barcelos-Neto and Wotzasek [1992b] Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. 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[2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982)
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D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982)
- Barcelos-Neto, J., Wotzasek, C.: Symplectic quantization of constrained systems. Mod. Phys. Lett. A 7, 1737–1748 (1992) https://doi.org/10.1142/S0217732392001439 Allys et al. [2016] Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982)
- Allys, E., Peter, P., Rodriguez, Y.: Generalized SU(2) Proca Theory. Phys. Rev. D 94(8), 084041 (2016) https://doi.org/10.1103/PhysRevD.94.084041 arXiv:1609.05870 [hep-th] Errasti Díez and Marinkovic [2022] Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982)
- Errasti Díez, V., Marinkovic, M.K.: Symplectic quantization of multifield generalized Proca electrodynamics. Phys. Rev. D 105(10), 105022 (2022) https://doi.org/10.1103/PhysRevD.105.105022 arXiv:2112.11477 [hep-th] Toms [2015] Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982)
- Toms, D.J.: Faddeev-Jackiw quantization and the path integral. Phys. Rev. D 92(10), 105026 (2015) https://doi.org/10.1103/PhysRevD.92.105026 arXiv:1508.07432 [hep-th] Dirac [1950] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982)
- Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950) https://doi.org/10.4153/CJM-1950-012-1 Dirac [2001] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982)
- Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, monograph series. Dover Publications, Berlin (2001) Rodrigues et al. [2018] Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982)
- Rodrigues, D.C., Galvão, M., Pinto-Neto, N.: Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach. Phys. Rev. D 98(10), 104019 (2018) https://doi.org/10.1103/PhysRevD.98.104019 arXiv:1808.06751 [gr-qc] Kamimura [1982] Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982) Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982)
- Kamimura, K.: SINGULAR LAGRANGIAN AND CONSTRAINED HAMILTONIAN SYSTEMS: GENERALIZED CANONICAL FORMALISM. Nuovo Cim. B 68, 22 (1982)
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